### Zorn's Lemma, The Well-Ordering Theorem, and Undefinability | Nathan Dalaklis

Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse ...

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**Zorn's lemma**, also known as the **Kuratowski–Zorn lemma**, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain necessarily contains at least one maximal element.

**Zorn's lemma**, also known as the **Kuratowski–Zorn lemma**, after mathematicians Max Zorn and Kazimierz Kuratowski, is a proposition of set theory that states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.

Proved by Kuratowski in 1922 and independently by Zorn in 1935, this lemma occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.

Zorn's lemma is equivalent to the well-ordering theorem and the axiom of choice, in the sense that any one of them, together with the Zermelo–Fraenkel axioms of set theory, is sufficient to prove the others. An earlier formulation of Zorn's lemma is Hausdorff's maximum principle which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.

Zorn's Lemma and The Well-ordering Theorem are seemingly straightforward statements, but they give incredibly mind-bending results. Orderings, Hasse ...

Functional Analysis by Prof. P.D. Srivastava, Department of Mathematics, IIT Kharagpur. For more details on NPTEL visit http://nptel.iitm.ac.in.

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